Regularity in the $\overline{\partial}$--Neumann problem, D'Angelo forms, and Diederich--Fornæss index
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arXiv
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2025
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| _version_ | 1866908319892897792 |
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| author | Straube, Emil J. |
| author_facet | Straube, Emil J. |
| contents | This article chronicles a development that started around 1990 with \cite{BoasStraube91}, where the authors showed that if a smooth bounded pseudoconvex domain $Ω$ in $\mathbb{C}^{n}$ admits a defining function that is plurisubharmonic at points of the boundary, then the $\overline{\partial}$--Neumann operators on $Ω$ preserve the Sobolev spaces $W^{s}_{(0,q)}(Ω)$, $s\geq 0$. The same authors then proved a further regularity result and made explicit the role of D'Angelo forms for regularity (\cite{BoasStraube93}). A few years later, Kohn (\cite{Kohn99}) initiated a quantitative study of the results in \cite{BoasStraube91} by relating the Sobolev level up to which regularity holds to the Diederich--Fornæss index of the domain. Many of these ideas were synthesized and developed further by Harrington (\cite{Harrington11,Harrington19,Harrington22}). Then, around 2020, Liu (\cite{Liu19b, Liu19}) and Yum (\cite{Yum21}) discovered that the DF--index is closely related to certain differential inequalities involving D'Angelo forms. This relationship in turn led to a recent new result which supports the conjecture that DF--index one should imply global regularity in the $\overline{\partial}$--Neumann problem (\cite{LiuStraube22}). Much of the work described above relies heavily on Kohn's groundbreaking contributions to the regularity theory of the $\overline{\partial}$--Neumann problem. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_03562 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Regularity in the $\overline{\partial}$--Neumann problem, D'Angelo forms, and Diederich--Fornæss index Straube, Emil J. Complex Variables 32W05, 32T99 This article chronicles a development that started around 1990 with \cite{BoasStraube91}, where the authors showed that if a smooth bounded pseudoconvex domain $Ω$ in $\mathbb{C}^{n}$ admits a defining function that is plurisubharmonic at points of the boundary, then the $\overline{\partial}$--Neumann operators on $Ω$ preserve the Sobolev spaces $W^{s}_{(0,q)}(Ω)$, $s\geq 0$. The same authors then proved a further regularity result and made explicit the role of D'Angelo forms for regularity (\cite{BoasStraube93}). A few years later, Kohn (\cite{Kohn99}) initiated a quantitative study of the results in \cite{BoasStraube91} by relating the Sobolev level up to which regularity holds to the Diederich--Fornæss index of the domain. Many of these ideas were synthesized and developed further by Harrington (\cite{Harrington11,Harrington19,Harrington22}). Then, around 2020, Liu (\cite{Liu19b, Liu19}) and Yum (\cite{Yum21}) discovered that the DF--index is closely related to certain differential inequalities involving D'Angelo forms. This relationship in turn led to a recent new result which supports the conjecture that DF--index one should imply global regularity in the $\overline{\partial}$--Neumann problem (\cite{LiuStraube22}). Much of the work described above relies heavily on Kohn's groundbreaking contributions to the regularity theory of the $\overline{\partial}$--Neumann problem. |
| title | Regularity in the $\overline{\partial}$--Neumann problem, D'Angelo forms, and Diederich--Fornæss index |
| topic | Complex Variables 32W05, 32T99 |
| url | https://arxiv.org/abs/2504.03562 |